Optimal. Leaf size=205 \[ \frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.517783, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3558, 3595, 3592, 3527, 3480, 206} \[ \frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3558
Rule 3595
Rule 3592
Rule 3527
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac{\int \frac{\tan ^3(c+d x) \left (-4 a+\frac{13}{2} i a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\int \frac{\tan ^2(c+d x) \left (-\frac{63 i a^2}{2}-\frac{141}{4} a^2 \tan (c+d x)\right )}{\sqrt{a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\int \tan (c+d x) \sqrt{a+i a \tan (c+d x)} \left (\frac{267 a^3}{2}-\frac{1083}{8} i a^3 \tan (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{\int \sqrt{a+i a \tan (c+d x)} \left (\frac{1083 i a^3}{8}+\frac{267}{2} a^3 \tan (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{i \int \sqrt{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}\\ \end{align*}
Mathematica [A] time = 1.84149, size = 149, normalized size = 0.73 \[ -\frac{e^{-4 i (c+d x)} \left (-33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}+1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{3/2} \sinh ^{-1}\left (e^{i (c+d x)}\right )+3\right )}{60 a^2 d \left (1+e^{2 i (c+d x)}\right )^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 131, normalized size = 0.6 \begin{align*} 2\,{\frac{1}{d{a}^{4}} \left ( 1/3\, \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{3/2}-3\,a\sqrt{a+ia\tan \left ( dx+c \right ) }-1/16\,{a}^{3/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+ia\tan \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) -{\frac{31\,{a}^{2}}{8\,\sqrt{a+ia\tan \left ( dx+c \right ) }}}+3/4\,{\frac{{a}^{3}}{ \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{3/2}}}-1/10\,{\frac{{a}^{4}}{ \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{5/2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44391, size = 1003, normalized size = 4.89 \begin{align*} -\frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (983 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 1527 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 348 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 33 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3\right )} e^{\left (i \, d x + i \, c\right )} + 15 \, \sqrt{\frac{1}{2}}{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \sqrt{\frac{1}{a^{5} d^{2}}} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 15 \, \sqrt{\frac{1}{2}}{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \sqrt{\frac{1}{a^{5} d^{2}}} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{120 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{5}{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{5}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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